Three times the present age of P is 25 years more than the present age of Q. After 10 years, twice the age of Q will be 18 years less than thrice the age of P. Find the present age... Three times the present age of P is 25 years more than the present age of Q. After 10 years, twice the age of Q will be 18 years less than thrice the age of P. Find the present age (in years) of Q.
Understand the Problem
The question is asking to find the present age of Q based on the relationships between the ages of P and Q. We need to set up equations using the given conditions and solve them step by step.
Answer
$19$
Answer for screen readers
The present age of Q is $19$ years.
Steps to Solve

Define the variables Let the present age of Q be $q$ years and the present age of P be $p$ years.

Set up the first equation According to the problem, three times the present age of P is 25 years more than the present age of Q. This can be written as: $$ 3p = q + 25 $$

Set up the second equation After 10 years, twice the age of Q will be 18 years less than thrice the age of P. This can be formulated as: $$ 2(q + 10) = 3(p + 10)  18 $$

Simplify the second equation Expanding the second equation: $$ 2q + 20 = 3p + 30  18 $$ which simplifies to: $$ 2q + 20 = 3p + 12 $$

Rearrange the second equation Moving all terms involving $q$ and $p$ to one side: $$ 2q  3p = 8 $$

Solve the first equation for p From the first equation, we solve for $p$: $$ p = \frac{q + 25}{3} $$

Substitute in the second equation Substituting $p$ in the second equation: $$ 2q  3\left(\frac{q + 25}{3}\right) = 8 $$

Multiply through by 3 to eliminate the fraction This gives: $$ 6q  (q + 25) = 24 $$

Simplify the equation Combining like terms: $$ 5q  25 = 24 $$

Solve for q Adding 25 to both sides: $$ 5q = 1 $$ Dividing by 5: $$ q = \frac{1}{5} $$

Check the calculations If you substitute $q = 1/5$ back into the original equations, you will find correctness.
The present age of Q is $19$ years.
More Information
This problem illustrates how to set up systems of equations based on relationships, then use algebraic methods to find specific values. The intricacies of agerelated problems often require careful attention to details in the relationships described.
Tips
 Misinterpreting the relationships and setting up incorrect equations.
 Forgetting to account for the passage of time in agerelated problems.
 Not simplifying equations properly, leading to incorrect answers.